This problem is a variation of coin change problem and can be solved in O(n) time and O(n) auxiliary space.

The idea is to create a table of size n+1 to store counts of all scores from 0 to n. For every possible move (3, 5 and 10), increment values in table.

C

// A C program to count number of possible ways to a given score
// can be reached in a game where a move can earn 3 or 5 or 10
#include <stdio.h>
// Returns number of ways to reach score n
int count(int n)
{
// table[i] will store count of solutions for
// value i.
int table[n+1], i;
// Initialize all table values as 0
memset(table, 0, sizeof(table));
// Base case (If given value is 0)
table[0] = 1;
// One by one consider given 3 moves and update the table[]
// values after the index greater than or equal to the
// value of the picked move
for (i=3; i<=n; i++)
table[i] += table[i-3];
for (i=5; i<=n; i++)
table[i] += table[i-5];
for (i=10; i<=n; i++)
table[i] += table[i-10];
return table[n];
}
// Driver program
int main(void)
{
int n = 20;
printf("Count for %d is %d\n", n, count(n));
n = 13;
printf("Count for %d is %d", n, count(n));
return 0;
}

Python3

# Python program to count number of possible ways to a given
# score can be reached in a game where a move can earn 3 or
# 5 or 10.
# Returns number of ways to reach score n.
def count(n):
# table[i] will store count of solutions for value i.
# Initialize all table values as 0.
table = [0 for i in range(n+1)]
# Base case (If given value is 0)
table[0] = 1
# One by one consider given 3 moves and update the
# table[] values after the index greater than or equal
# to the value of the picked move.
for i in range(3, n+1):
table[i] += table[i-3]
for i in range(5, n+1):
table[i] += table[i-5]
for i in range(10, n+1):
table[i] += table[i-10]
return table[n]
# Driver Program
n = 20
print('Count for', n, 'is', count(n))
n = 13
print('Count for', n, 'is', count(n))
# This code is contributed by Soumen Ghosh

Output:

Count for 20 is 4
Count for 13 is 2

Exercise: How to count score when (10, 5, 5), (5, 5, 10) and (5, 10, 5) are considered as different sequences of moves. Similarly, (5, 3, 3), (3, 5, 3) and (3, 3, 5) are considered different.

This article is contributed by Rajeev Arora. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above